ELECTROSTATIC WAVES IN PLASMAS....with reference to Landau Damping

A thesis submitted by

Kushal Kumar Shah

EE01B040

in partial fulfillment for the award of the degree of

BACHELOR OF TECHNOLOGY

under the esteemed guidance of

Dr. Hari Ramachandran

CERTIFICATE

This is to certify that the work contained in the thesis entitled “Electrostatic

Waves in Plasmas” submitted by Kushal Kumar Shah (EE01B040) in partial

fulfillment of the requirements for the award of the degree of Bachelor of Tech-

nology in Electrical Engineering is a bona fide work carried out by him under my

guidance and supervision.

Project Guide

Dr. Hari Ramachandran

Associate Professor

Department of Electrical Engineering

Indian Institute of Technology Madras

Place: Chennai

Date:

i

Acknowledgments

At times our own light goes out and is rekindled by a spark from another person.

Each of us has cause to think with deep gratitude of those who have lighted the

flame within us.

— Albert Schweitzer

My one year long association with my B.Tech project guide, Dr. Hari Ra-

machandran, has been a wonderful learning experience. I would like to extend my

heartfelt gratitude to Dr. Ramachandran for his help, guidance and the patience

with which he taught me the intricate but interesting subject of Plasma Physics.

This project has been the most enjoyable part of my academic life at IIT Madras.

I will be ever obliged to IIT Madras in general and the Electrical Engineering

Department in particular for having provided me with such an intellectually stim-

ulating environment that is conducive to high academic growth. It was only due

to the flexible system of IIT Madras that I was able to do my final year project in

an unconventional area like Plasma Physics.

I am thankful to all my professors at IIT Madras who taught me the basic prin-

ciples of topics like Laplace transform, Fourier transform, differential equations,

electromagnetic fields, to name a few, without which it would have been impossible

for me to do research in the area of Plasma Physics.

ii

Finally, I would like to thank my parents, Mr. and Mrs. Kishan Shah, for the

motivation that they have provided me throughout my academic career in general

and the past one year in particular.

Kushal Kumar Shah

May, 2005

iii

Abstract

This thesis considers the effects of an electrostatic wave on a pure electron one

dimensional plasma where the ions are essentially at rest. This is primarily a

numerical analysis. We begin by trying to understand what happens to the zero

order Maxwellian density distribution function in the presence of a landau damped

sinusoidal electrostatic field. This is for the case of a homogeneous plasma. Then

we turn our attention to an initially inhomogeneous plasma. The electrostatic field

is no longer sinusoidal. We come across what is called the airy function. Now,

we add some noise to the plasma in the form of a random phased spectrum of

waves. So, instead of a single airy function, our electrostatic field is now described

by a sum of shifted versions of multiple airy functions with random phase. In the

end, we allow our electrostatic field to be modified due to changes in the density

distribution function itself. The field is airy only as long as the density gradient

is linear. But as the particles move around, the linearity is no longer maintained.

We were able to reach a steady state where the field and the distribution function

were self-consistent. One of the graphs obtained from the simulation also bears

the fingerprints of landau damping.

iv

Contents

I THEORY 1

1 Introduction 2

2 Homogeneous Plasmas 7

3 Inhomogeneous Plasmas 10

3.1 Single Airy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Multiple Airy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Wave-particle interaction . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Conclusion 37

A dist.f 39

B path.ai.f 45

C path.va.f 49

D Green’s function technique 54

E feedback.f 57

v

List of Figures

1.1 Statistical Description of a Plasma . . . . . . . . . . . . . . . . . . 2

1.2 Small oscillations in plasma density . . . . . . . . . . . . . . . . . . 3

1.3 Flattening of the Gaussian distribution function due to landau damp-

ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Trajectory of a trapped particle in phase space in the reference frame

in which the wave is at rest. . . . . . . . . . . . . . . . . . . . . . . 4

1.5 The airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 This is the plot of the trajectory of a trapped particle for the case

of a homogeneous plasma. This is in the wave frame. . . . . . . . . 8

2.2 This is the plot of the trajectory of a trapped particle for the case

of a homogeneous plasma. This is in the wave frame. Unlike the

previous figure, in this one, the particle is trapped only for a short

time and then it gets free. . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Modified distribution function for homogeneous plasma . . . . . . . 9

2.4 This is the zoomed-in version of the previous figure. . . . . . . . . . 9

3.1 This is the electrostatic field when modeled as a single airy function. 13

3.2 Trapped particle orbit for single airy field . . . . . . . . . . . . . . . 15

3.3 A particle orbit for single airy field . . . . . . . . . . . . . . . . . . 15

vi

3.4 Plot of the final relative movements of particles for the case of in-

homogeneous plasma where the electrostatic wave is a single airy

function. This is for a very weak field . . . . . . . . . . . . . . . . . 16

3.5 Plot of the final relative movements of particles for the case of in-

homogeneous plasma where the electrostatic wave is a single airy

function. This is for intermediate field strengths . . . . . . . . . . . 17

3.6 Plot of the final relative movements of particles for the case of in-

homogeneous plasma where the electrostatic wave is a single airy

function. This is for a very strong field . . . . . . . . . . . . . . . . 18

3.7 This is the electrostatic field when modeled as multiple airy func-

tions at t=0. The horizontal axis is the z-coordinate. . . . . . . . . 20

3.8 This is the magnitude of the space-Fourier transform of the electro-

static field when modeled as multiple airy functions at t=0. . . . . . 20

3.9 This is the phase of the space-Fourier transform of the electrostatic

field when modeled as multiple airy functions at t=0. It can be seen

that the shifted airy functions are essentially out of phase. . . . . . 21

3.10 This is the electrostatic field when modeled as multiple airy func-

tions at t=1.2. The horizontal axis is the z-coordinate. . . . . . . . 21

3.11 This is the magnitude of the space-Fourier transform of the electro-

static field when modeled as multiple airy functions at t=1.2. . . . . 22

3.12 Phase of the space-Fourier transform of the electrostatic field when

modeled as multiple airy functions at t=1.2 . . . . . . . . . . . . . 22

3.13 Plot of the trajectory of one of the particles when the electrostatic

field is a sum of multiple airy functions . . . . . . . . . . . . . . . . 23

vii

3.14 Plot of the trajectory of one of the particles when the electrostatic

field is a sum of multiple airy functions . . . . . . . . . . . . . . . . 23

3.15 Plot of the final distribution of particles for the case of inhomoge-

neous plasma where the electrostatic wave is modeled as a sum of

multiple airy functions. This is for the case of a very weak electric

field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.16 Plot of the final distribution of particles for the case of inhomoge-

neous plasma where the electrostatic wave is modeled as a sum of

multiple airy functions. This is for the case of a very strong electric

field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.17 This is the 1st order electric field at t=100 for the case of wave-

particle interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.18 This is the density distribution function at t=100 for the case of

wave-particle interaction . . . . . . . . . . . . . . . . . . . . . . . . 29

3.19 This shows the in-phase component of the density perturbation.

NC =∑t=t2t=t1 N(t) cos(wt). . . . . . . . . . . . . . . . . . . . . . . . 30

3.20 This shows the quadrature component of the density perturbation.

NQ =∑t=t2t=t1 N(t) sin(wt). . . . . . . . . . . . . . . . . . . . . . . . 30

3.21 This plots the total energy of all the particles in the system as a

function of time for the case of wave-particle interaction . . . . . . 31

3.22 This is the plot of the energy of the system as a function of z at

t=100 for the case of wave-particle interaction . . . . . . . . . . . . 31

3.23 Plot of the paths of particles in the lab frame selected from all over

the velocity range. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

viii

3.24 Plot of the path of a particle in the wave frame. This particle is

initially free, then gets trapped and at the end gets free again. . . 32

3.25 This plot shows the relative distribution of particles in various ve-

locity bins for the case of wave-particle interaction . . . . . . . . . . 33

3.26 This is the contour plot of the magnitude of the density perturba-

tions in phase space for the case of wave-particle interaction . . . . 36

D.1 The airy function, Ai(-z) . . . . . . . . . . . . . . . . . . . . . . . . 55

D.2 The airy function, Bi(-z) . . . . . . . . . . . . . . . . . . . . . . . . 56

ix

List of Tables

3.1 Simulation results for the case of an inhomogeneous plasma when

the electric field is modeled as a single airy function. . . . . . . . . 14

3.2 Simulation results for the case of an inhomogeneous plasma when

the electric field is modeled as a sum of multiple airy functions. . . 19

x

Part I

THEORY

1

Chapter 1

Introduction

In this thesis we present our study of electrostatic waves in homogeneous as well

as inhomogeneous plasma with reference to the phenomenon of landau damping.

Figure 1.1: Statistical description of a plasma. The random arrows represent thethermal velocity of the particles. The velocity distribution of the particles is fairlyMaxwellian. But the characteristics of this Maxwellian are different in differentparts of the plasma. The electron and ion temperature can be different in differentparts of the plasma.

Figure 1.1 gives a statistical description of a plasma. The random arrows repre-

sent the thermal velocity of the particles. The velocity distribution of the particles

2

is fairly Maxwellian. But the characteristics of this Maxwellian are different in

different parts of the plasma. The electron and ion temperature can be differ-

ent in different parts of the plasma. An electron or ion can even have different

temperatures along the three co-ordinate axes. For all our present analysis we

have considered a one-dimensional warm electron plasma in which the ions are

essentially at rest.

Figure 1.2: Small oscillations in plasma density. This graph shows an arbitrarybackground density distribution function on which a sinusoidal density perturba-tion is superimposed.

Figure 1.2 depicts the small oscillations in the plasma density superimposed on

the background electron density.

Figure 1.3 depicts the effect of landau damping on the velocity dependent

distribution function of a plasma. It can be seen that a small region of the Gaussian

curve has been flattened. This happens near the region where the velocity of the

electrons is close to the phase velocity of the wave. This is primarily due to particle

trapping and phase mixing. Figure 1.4 shows what we mean by particle trapping.

In the field of plasma physics, the term “landau damping” is used to describe

two separate damping mechanisms. One is the resonance damping of a wave and

3

Figure 1.3: Flattening of the Gaussian distribution function due to landau damp-ing.

Figure 1.4: Trajectory of a trapped particle in phase space in the reference framein which the wave is at rest.

the second is the damping of beat waves. We use the term landau damping to

describe the resonance damping of waves. This kind of damping occurs mainly due

to energy transfer from the wave to the particles that are moving at speeds close

to the phase velocity of the wave. It must however be noted that this resonance

interaction can also cause growth which happens when the number of particles

with velocity slightly higher than the phase velocity of the wave is more than the

4

particles with a slightly lower velocity. The particles traveling with velocity away

from the phase velocity of the wave do not get involved in the damping or growth

mechanisms.

For an initially homogeneous plasma, landau damping of electrostatic waves[1]

is understood to quite a large extent. But for inhomogeneous plasmas, the phe-

nomenon is highly non-trivial. Considerable amount of work has been done by

researchers all around the world in this area but still we do not have a very clear

picture of what exactly happens when we have non-uniformities in the plasma.

Although we were not able to get a perfect understanding of the underlying phe-

nomenon, the insights derived from our work may be useful for further research

on the topic.

The homogeneous case can be analyzed by usage of the Vlasov equation coupled

with the Poisson’s equation. But for the inhomogeneous case, we have to content

ourselves with the fluid equations. There are a few publications that have dealt

with inhomogeneous plasmas using the Vlasov equation but those are beyond the

scope of our present work. But even those few publications are more like works of

mathematics with a lot of assumptions. The physics remains obscure.

First order perturbation theory and linearization form the common theme of

our mathematical methods. Since we do not have exact analytical expressions for

describing the reaction of the plasma to the electrostatic waves in homogeneous as

well as inhomogeneous plasmas, we have to fall back on simulations to get a feel

of the physics involved in the process.

This thesis presents our work in increasing order of complexity. First, we

consider the case of a homogeneous plasma. Then, we go on to inhomogeneous

plasma where we first consider the case of the electrostatic field modeled as a single

5

airy function (Figure 1.5).

Figure 1.5: The airy function. The frequency of time oscillations, w, of this waveis equal to the plasma frequency, we, at z=0. Hence, the wave has a cut-off at thispoint.

Then, we discuss the case of a noisy plasma. But for all the above cases we

do not consider the nonlinear modification of the electric field itself due to density

perturbations. In the last section, we consider this effect and try to understand

how the electric field would react to changes in the zero-order plasma density.

All simulations were done in Fortran 77. For solving differential equations,

doing integrations and most of the purely mathematical things, use was made of

the codes given in Numerical Recipes in Fortran 77.

6

Chapter 2

Homogeneous Plasmas

In this chapter we discuss the phenomenon of landau damping for the case of a

homogeneous plasma[1]. We have used the Vlasov and the Poisson’s equations for

solving the problem. Analytical results pertaining to the problem can be obtained

from Nicholson[2].

Since the plasma is homogeneous, we can Fourier transform the equations in

time as well as space. To get landau damping, we must consider the problem

as an initial value problem. Now, when we have damping, the frequency of time

oscillations is no longer real. It becomes complex. We use this information for

simulating the particle trajectories. The electrostatic field is modeled as an expo-

nentially decaying sinusoidal function. The initial distribution function is assumed

to be Maxwellian.

It was found that particles with initial velocity lower than the phase velocity of

the wave were speeded up and vice versa. The particles with velocity far away from

the phase velocity of the wave do not get involved in the damping mechanisms. We

also see a very interesting phenomenon called particle trapping. The trajectories

of two such trapped particles can be seen in Figures 2.1 and 2.2.

7

Figure 2.1: This is the plot of the trajectory of a trapped particle for the case ofa homogeneous plasma. This is in the wave frame.

Figure 2.2: This is the plot of the trajectory of a trapped particle for the case ofa homogeneous plasma. This is in the wave frame. Unlike the previous figure, inthis one, the particle is trapped only for a short time and then it gets free.

Some of the particles get essentially trapped in the wave. If the wave were not

damped, these particles would be forever trapped. But since we have damping,

the trapped particles happen to escape out at some point of time. It is these

trapped particles that are responsible for wave damping. In the initial stages, the

damping is linear. But when the trapped particles turn around to complete their

first circular trajectory, the damping enters its nonlinear phase. The nonlinear

8

part of landau damping is due to phase mixing.

The initial and the modified distribution functions are plotted in Figures 2.3

and 2.4.

Figure 2.3: This is the plot of the original and the modified distribution functionfor the case of a homogeneous plasma. The horizontal axis is the velocity and thevertical axis is the number of particles. The velocity v=0 represents the resonancepoint. This is in the wave frame. It can be clearly seen that the particles thatwere initially on the left of v=0 have now moved to the right.

Figure 2.4: This is the zoomed-in version of the previous figure.

9

Chapter 3

Inhomogeneous Plasmas

To analyze the nature of electrostatic wave propagation in nonuniform plasmas, we

have used the fluid equations coupled with the Poisson’s equation. Let us forget

landau damping for a while.

We have modeled our non uniformity as a linear density profile. The electro-

static wave turns out to be an airy function. The derivation of the airy equation

is given below.

We have used the 1st order perturbation theory to solve for the electric field.

The density is modeled as n = n0 +n1 and similarly for velocity and electric field.

Here n0 represents the zero order quantity and n1 is the small perturbation in the

zero order value. These quantities are described below.

The zero-oder quantities are,

n0 = n0(z) Density (3.1)

v0 = 0 Velocity (3.2)

E0 = 0 Electrostatic Field (3.3)

10

The 1st-oder quantities are,

n1 = n1(z) exp(−iwt) Density (3.4)

v1 = v1(z) exp(−iwt) Velocity (3.5)

E1 = E1(z) exp(−iwt) Electrostatic Field (3.6)

The only equations used to derive the results are the fluid equations and the

Poisson’s equation.

1. Continuity equation:

∂n

∂t+ ~∇ · (n~v) = 0 (3.7)

2. Equation of motion:

men∂~v

∂t+men(~v · ~∇)~v = nqe ~E + nqe(~v × ~B)− ~∇p (3.8)

3. Poisson’s equation:

~∇ · ~E =qen

εo(3.9)

The zero-order continuity equation is satisfied easily. To satisfy the zeroth-

order momentum equation we introduce a term D that takes care of the ~∇p term.

This D can be attributed to the volume rate of the recombination, ionization, and

loss processes in equilibrium[3].

Now, let us write the 1st order equations. We can use the fact that pn−γ =

constant.

11

1. Continuity equation:

∂n1

∂t+∂(n0v1)

∂z= 0 (3.10)

2. Equation of motion:

men0∂v1

∂t= −n0eE1 − γKT

∂n1

∂z(3.11)

3. Poisson’s equation:

∂E1

∂z= −en1

εo(3.12)

In writing the above 1st order equations, we have neglected the following terms,

• n1v1 in the continuity equation

• n1∂v1

/∂t, v1∂v1

/∂z, n1E1, in the momentum equation

Now, substituting for n1, v1 and E1, in the momentum equation, we get,

−iwv1(z) =−eme

E1(z)− γKT

men0(z)

∂n1

∂z(3.13)

v1(z) =i

w

[−eme

E1(z)− γKT

men0(z)

∂n1(z)

∂z

](3.14)

Substituting this in the first order continuity equation, we have,

wn1 +n0e

wme

∂E1

∂z+γKT

mew

∂2n1

∂z2+

e

mew

∂n0

∂zE1 = 0 (3.15)

12

Now, substituting the Poisson’s equation, ∂E1

∂z= −en1/ε0, and n0(z) = n0(1−

z/L) into the above equation, we have,

v2e

w2

∂3E1

∂z3+[1− w2

e

w2(1− z/L)

]∂E1

∂z+

w2e

w2LE1 = 0 (3.16)

In the above equation, v2e = γKT

/me, w

2e = e2n0

/meε0, and L(z << L) is the

scale length of the plasma. If we write b = w2e

/v2eL and use the fact that w = we,

then, we can write the above equation as,

∂3E1

∂z3+ bz

∂E1

∂z+ bE1 = 0 (3.17)

∂3E1

∂z3+ b(z

∂E1

∂z+ E1) = 0 (3.18)

∂3E1

∂z3+ b

∂zE1

∂z= 0 (3.19)

Integrating the above equation w.r.t z, we get,

∂2E1

∂z2+ bzE1 = 0 (3.20)

The above is the well known airy equation. This function has been plotted in

Figure 3.1.

Figure 3.1: This is the electrostatic field when modeled as a single airy function.

13

3.1 Single Airy

This section considers the case where the electrostatic field is modeled as a single

airy function. This is the simplest possible case for an inhomogeneous plasma. So

far we have been able to find an equation for the electrostatic wave in a plasma

with a linear density profile. But this is not the end of the road. We still do not

know the effect of such an electric field on the plasma particles. This is where we

leave the world of mathematics and enter the world of simulations. It should be

noted that as the particles move about, the density profile does not remain linear.

So, in reality, the electrostatic wave no longer remains airy. But, for the time

being, we have neglected the modifications of the wave profile due to the motion

of particles. We will take up this issue in a later section.

The results of the simulation are as follows:

z-range v-range endright (%)0 to 10 -1.0 to 1.0 55.340 to 10 -2.0 to 2.0 57.370 to 10 -3.0 to 3.0 56.400 to 10 -3.5 to 3.5 55.670 to 10 -10.0 to 10.0 74.76

-10 to 10 -3.5 to 3.5 46.75

Table 3.1: Simulation results for the case of an inhomogeneous plasma when theelectric field is modeled as a single airy function.

In Table 3.1, the z-range and the v-range are normalized values. endright is the

percentage of particles that end up on the right side of the gradient i.e. towards

the low density side. This can be understood better from Figures 3.4, 3.5 and 3.6

It can be seen from the above table that the movement of the particles is

essentially diffusive in nature and the electric field tends to smoothen the density

gradient.

14

For a better feel of particle trajectories please view Figures 3.2 and 3.3

Figure 3.2: Plot of the trajectory of one of the trapped particles when the elec-trostatic field is a single airy function. We can see that this particle is trappedin the airy wave and will most probably remain so for ever. What is even moreinteresting is that it keeps on following the same path in phase space in everycycle. This is not so common.

Figure 3.3: Plot of the trajectory of one of the particles when the electrostaticfield is a single airy function. This particle gets trapped initially for a while buteventually frees itself.

15

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-8 -6 -4 -2 0 2 4 6 8 10

v

z

"tl.dat" u 2:1"tr.dat" u 2:1

Figure 3.4: Plot of the final relative movements of particles for the case of inho-mogeneous plasma where the electrostatic wave is a single airy function. This isfor a very weak field. The × dots represent the particles that went to the left buteventually turned around and ended up on the right. The + dots represent theparticles that went to the right but eventually turned around and ended up onthe left. Here, right represents the low density side. It must also be noted thatthe dots also include those particles that turned around more than once. We cansee that this graph shows much more structure when compared to Figures 3.5 and3.6. Here the trapping happens only at certain discrete values of velocities, whichcan be inferred from the horizontal lines in the graph. But we do not clearly seea separate trapping velocity range. This is mainly because the electric field is notlow enough.

16

-1

-0.5

0

0.5

1

1.5

-10 -8 -6 -4 -2 0 2 4 6 8 10

v

z

"tl.dat" u 2:1"tr.dat" u 2:1

Figure 3.5: Plot of the final relative movements of particles for the case of inhomo-geneous plasma where the electrostatic wave is a single airy function. This is forintermediate field strengths. The × dots represent the particles that went to theleft but eventually turned around and ended up on the right. The + dots representthe particles that went to the right but eventually turned around and ended up onthe left. Here, right represents the low density side. It must also be noted that thedots also include those particles that turned around more than once. This graphhas a lot less structure when compared to Figure 3.4. We can see some horizontallines but there are a lot of loops also formed. It is not very clear what these loopsrepresent. These loops are certainly caused by the high nonlinearity involved inthe simulation.

17

-1.5

-1

-0.5

0

0.5

1

1.5

2

-10 -8 -6 -4 -2 0 2 4 6 8 10

v

z

"tl.dat" u 2:1"tr.dat" u 2:1

Figure 3.6: Plot of the final relative movements of particles for the case of inho-mogeneous plasma where the electrostatic wave is a single airy function. This isfor a very strong field. The × dots represent the particles that went to the leftbut eventually turned around and ended up on the right. The + dots representthe particles that went to the right but eventually turned around and ended upon the left. Here, right represents the low density side. It must also be noted thatthe dots also include those particles that turned around more than once. We canclearly see from Figures 3.4, 3.5 and the above that the number of particles thatturn around in their trajectories goes up with the field intensity. It is very difficultto find any definite pattern in this graph. It is a highly nonlinear situation.

18

3.2 Multiple Airy

Now, we add some noise to the plasma in the form of a random phased spectrum

of waves. In this case, the electrostatic field is essentially a sum of a large number

of shifted airy functions with random phase. The electric field at t=0 is plotted

as a function of the space in Figure 3.7. It is interesting to note that although the

starting phases are random, at some points of time the individual airy functions

do get tuned in. The simulations are carried out as in the previous section. The

results of the simulation are:

v-range endright (%)-1.0 to 1.0 20.41-2.0 to 2.0 17.86-3.0 to 3.0 19.52-3.5 to 3.5 10.40

-10.0 to 10.0 35.89

Table 3.2: Simulation results for the case of an inhomogeneous plasma when theelectric field is modeled as a sum of multiple airy functions.

For all the above results, the z-range is -10.0 to 10.0. In Table 3.2, the z-range

and the v-range are normalized values. endright is the percentage of particles that

end up on the right side of the gradient i.e. towards the low density side. This

can be understood better by having a look at Figures 3.15 and 3.16.

It can be seen that this case is very different from the earlier case of a single

airy field. Now, majority of the particles are moving up the density gradient. Two

of the particle trajectories can be viewed in Figures 3.13 and 3.14

19

Figure 3.7: This is the electrostatic field when modeled as multiple airy functionsat t=0. The horizontal axis is the z-coordinate.

Figure 3.8: This is the magnitude of the space-Fourier transform of the electrostaticfield when modeled as multiple airy functions at t=0.

20

Figure 3.9: This is the phase of the space-Fourier transform of the electrostaticfield when modeled as multiple airy functions at t=0. It can be seen that theshifted airy functions are essentially out of phase.

Figure 3.10: This is the electrostatic field when modeled as multiple airy functionsat t=1.2. The horizontal axis is the z-coordinate.

21

Figure 3.11: This is the magnitude of the space-Fourier transform of the electro-static field when modeled as multiple airy functions at t=1.2.

Figure 3.12: This is the phase of the space-Fourier transform of the electrostaticfield when modeled as multiple airy functions at t=1.2. This is an interesting graphsince it shows that at some instances of time the initially random airy functionsdo get tuned for a short time.

22

Figure 3.13: Plot of the trajectory of one of the particles when the electrostatic fieldis a sum of multiple airy functions. This is for the case of a noisy inhomogeneousplasma. This plot shows that the particle was trapped for a short time but beforeit could complete a cycle in phase space, it got detrapped.

Figure 3.14: Plot of the trajectory of one of the particles when the electrostatic fieldis a sum of multiple airy functions. This is for the case of a noisy inhomogeneousplasma. This is a very interesting plot as it shows a particle that is undergoingactive trapping and detrapping. We can see that the particle was trapped initiallyand then it got detrapped. But unlike other particles, this one managed to gettrapped again at another location in phase space and got detrapped once again.

23

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-10 -8 -6 -4 -2 0 2 4 6 8 10

v

z

"tl.dat" u 2:1"tr.dat" u 2:1

Figure 3.15: Plot of the final distribution of particles for the case of inhomogeneousplasma where the electrostatic wave is modeled as a sum of multiple airy functions.This is for the case of a very weak electric field. The vertical axis represents thex-range. The horizontal axis represents the v-range. The × dots represent theparticles that went to the left but eventually turned around and ended up on theright. The + dots represent the particles that went to the right but eventuallyturned around and ended up on the left. Here, right represents the low densityside. It must also be noted that the dots also include those particles that turnedaround more than once. This graph shows a structure that is similar to Figure3.4. We can see that most of the trapping at non zero velocities occurs aroundv = ±0.2. There are a lot of particles getting trapped at v = 0 but that is not thevelocity range we are interested in.

24

Figure 3.16: Plot of the final distribution of particles for the case of inhomogeneousplasma where the electrostatic wave is modeled as a sum of multiple airy functions.This is for the case of a very strong electric field. The vertical axis represents thex-range. The horizontal axis represents the v-range. The × dots represent theparticles that went to the left but eventually turned around and ended up on theright. The + dots represent the particles that went to the right but eventuallyturned around and ended up on the left. Here, right represents the low densityside. It must also be noted that the dots also include those particles that turnedaround more than once. This plot is too cluttered to make any sense out of it.This is a highly nonlinear problem and is out of the scope of the present work.

25

3.3 Wave-particle interaction

So far, we have not considered the modification of the electrostatic field itself due

to modification in the density profile. In this section, we consider the linear effects

of the perturbation in density on the first order electrostatic field, which is a single

airy function to begin with. Also, in the previous two sections, we only considered

the paths of the particles. We did not really estimate the modified density profile.

But in this case, it is mandatory to calculate the modified density profile because

this is the input to calculate the new electric field.

Thus, it is mandatory to find a mechanism that would sustain the gradient

in the density profile in the absence of the airy field. We do this by introducing

an imbalance in the flux of particles entering the plasma on either extremes, a

nonuniform zero order electric field, E0, and collisions.

The imbalance in the flux is created in the following way. As soon as a particle

leaves the system at z1, which is the leftmost extreme of the plasma, we introduce

a new particle at z2, the rightmost extreme but with a weight, Aminus. And

similarly when a particle leaves the system at z2, we introduce a new particle at

z1, with weight, Aplus. And, Aplus > Aminus. These weights multiply with the

Maxwellian to give the actual number of particles that each orbit represents. Thus

we have more particles entering z1 compared to z2.

E0(z) is derived from the momentum equation. The derivation is as follows.

The zero order quantities are:

n0 = n0(z) Density (3.21)

v0 = 0 Velocity (3.22)

26

E0 = E0(z) Electrostatic Field (3.23)

The momentum equation is,

men∂~v

∂t+men(~v · ~∇)~v = nqe ~E + nqe(~v × ~B)− ~∇p (3.24)

Retaining only the zero order quantities, we get,

−ne ~E − ~∇p = 0 (3.25)

We know that,

~∇p = γKT∂n

∂z(3.26)

Our density gradient is linear. Thus,

n = n0(1− z

L) (3.27)

where n0 is the density at z = 0 and L is the scale length of our plasma.

Substituting these in the momentum equation, we get,

−eE0

m=−v2

e

L− z (3.28)

where ve =

√γKT

/m is the thermal velocity of the particles.

To model collisions we have used probabilistic swapping of the sign of particle

velocities. This means that after calculation of the location of each particle in the

phase space after each time instant, we generate a random number, pr (say), and

if this number is lesser than p times the absolute value of the particle velocity,

then we swap the sign of the velocity. The value of p is set by the user and this

27

controls the amount of collisions. A larger value of p implies more collisions. We

have run our simulations for a value of p = 0.01.

It was seen that the gradient is maintained to a fairly large extent. There are

spikes at points, but that does not cause much trouble.

Now, we add the airy field to our problem. The modified version of the airy

equation for this case is,

∂2E1

∂z2+w2ez

v2eLE1 =

w2en1

v2en0(0)

E0 (3.29)

To solve for E1 we use the green’s function technique (please refer page 54 for

further discussion on this topic). The solutions of the homogeneous equation are

the inverted versions of Ai and Bi, which are the standard airy functions. Thus,

E1 = cAi(-z)+Bi(-z)∫ z

−∞Ai(-z)

[ w2en1

v2en0(0)

E0(z)]dz+Ai(-z)

∫ ∞

zBi(-z)

[ w2en1

v2en0(0)

E0(z)]dz

(3.30)

where c is an arbitrary constant. The choice of c depends on the amount

of nonlinearity one wants to inculcate in the problem. As c goes on increasing

the problem becomes more and more nonlinear. We are interested in a weakly

nonlinear problem and we want the trapping velocity to be on the tail of the

Maxwellian.

The simulation is run for 1000 iterations and the results are shown in the

plots that follow. Since we have considered a very weak coupling of the wave and

the density perturbation, the airy field does not get modified much except at the

extreme left of the plasma as can be seen in Figure 3.17.

28

Figure 3.17: This is the 1st order electric field at t=100 for the case of wave-particleinteraction. We can see that most of the field is unmodified. Only the first fewvalues of the field are modified due to the density perturbation. This is mainlybecause the density distribution function itself maintains a gradient on an averageeven at t=100. If the density function itself shows wide variations from what itwas at t=0, then the electric field will also get largely modified.

Figure 3.18: This is the density distribution function at t=100 for the case ofwave-particle interaction. It can be seen that on an average we still have a lineargradient. Initially the density distribution does show variations from linearity butas time goes on the function settles down to structure shown in this graph.

29

Figure 3.19: This shows the in-phase component of the density perturbation.NC =

∑t=t2t=t1 N(t) cos(wt).

Figure 3.20: This shows the quadrature component of the density perturbation.NQ =

∑t=t2t=t1 N(t) sin(wt).

30

Figure 3.21: This plots the total energy of all the particles in the system as afunction of time for the case of wave-particle interaction. We can see that totalenergy of the system is decreases by around 2.3% from its initial value of 1.695and then stablizes to a value of around 1.655. The values shown are scaled valuesand should be taken in the relative sense.

Figure 3.22: This is the plot of the energy of the system as a function of z at t=100for the case of wave-particle interaction. We can clearly see the gradient in theenergy of the system.

31

Figure 3.23: Plot of the paths of particles in the lab frame selected from all overthe velocity range.

Figure 3.24: Plot of the path of a particle in the wave frame. This particle isinitially free, then gets trapped and at the end gets free again.

32

Figure 3.25: This plot shows the relative distribution of particles in various velocitybins. This is the magnitude of the perturbation and the underlying Gaussian hasbeen eliminated.NC =

∑t=t2t=t1 N(t) cos(wt)

NQ =∑t=t2t=t1 N(t) sin(wt)

NbinCQ =√NC2 +NQ2

We can clearly see peaks near v = −0.2 and v = +0.2. These velocities are thetrapping velocities. We know that due to landau damping the number of particlesslightly faster than the phase velocity of the wave increases. It is this phenomenonthat has been captured by the above plot. We can also see a peak near v = 0.05.This is because there is a lot of wave activity in that region and due to this eventhe particles with fairly low velocities are getting trapped.

33

The contour plot of the magnitude of the density perturbations in phase space

is shown in figure 3.26 on page 36. The underlying Gaussian distribution function

has been eliminated. It is this plot that gives us the maximum amount of informa-

tion about our simulation. To understand the story that this plot tells, we have

marked the interesting regions by labels, namely, 1,2,3,..., 11. Each of these labels

corresponds to a region that has a characteristic of its own. Some of the labels can

be grouped and seen together to understand yet another phenomenon.

Regions 1, 2, and 3 represent low velocity particles that have been trapped

in the airy field. The velocities of these particles is below the thermal velocity,

ve = 0.07. It is strange that these particles are getting trapped. We can attribute

this trapping region to the nonlinear effects of the wave. If we magnify this plot, we

will be able to see that these regions when put together closely follows a inverse

square root curve in phase space. This is also expected because the resonance

velocity is a function of the phase velocity of the electric field wave and that in

turn is a function of the square root of the density gradient. Thus the resonance

velocity goes like√

1− z/L, where L is the scale length of the density gradient of

the plasma. And this is what is shown by these regions.

Regions 4 and 5 represent particles that are in the tail of our Gaussian distri-

bution. We had chosen the magnitude of our airy field such as to get trapping in

the tail region of the Maxwellian. Ideally this region should have been in between

velocity range of v = −1.9 and v = −2.1. But instead we see this region slightly

beyond v = −2.1. This, however, is nothing to be concerned about because the

estimation of the trapping region involves a lot of approximations and slight de-

viations are understandable. We see some interesting contours immediately above

regions 4 and 5. We shall discuss them later when we discuss regions 9, 10 and 11.

34

Regions 6, 7 and 8 represent the two most uninteresting areas in our contour

plot. We do see some peaks in regions 6 and 8 but these are scattered around and

show no structure. Region 7 shows no contours at all. In this region the initial

density of particles is very high because this region corresponds to the peak of the

Maxwellian. Also, the electric field wave in the region, z < 0, is very negligible.

So, all that these particles see is the background zero order electric field, E0. This

electric field, E0(z), is such that it maintains the gradient. So, these particles do

not move much because their velocities are also very low. And we have clipped off

the very high values of density while generating this plot.

Regions 9, 10 and 11 again show some trapping phenomenon. We can see a

certain symmetry between the contours of this region and that of regions 4 and 5.

Even here the trapping occurs for velocities slightly higher than v = 2.1. Now let

come to the contours immediately above regions 4 and 5. We see similar patterns

below regions 9, 10 and 11. We also see that the number of these contours decreases

with increasing z. These contours represent slightly trapped particles that are

diffused over a larger velocity range. The range of velocities for which particles are

trapped depends on the magnitude of the electric field wave. And since we have

a airy field and not a sinusoidal field, the magnitude of the space dependent field

keeps on decreasing with increasing z. Thus the velocity range of trapping is more

near z = 0 and less at z = 10. And that is what we are seeing in our plot.

35

Figure 3.26: This is the contour plot of the magnitude of the density pertur-bations in phase space. The underlying Gaussian distribution function has beeneliminated. It is this plot that gives us the maximum amount of information aboutour simulation. To understand the story that this plot tells, we have marked theinteresting regions by labels, namely, 1,2,3,..., 11. Each of these labels correspondsto a region that has a characteristic of its own. Some of the labels can be groupedand seen together to understand yet another phenomenon.

36

Chapter 4

Conclusion

In this thesis we have studied the behavior of electrostatic waves in both ho-

mogeneous and inhomogeneous plasmas. We have also seen effects like Landau

damping for both these cases. For a homogeneous plasma, the phenomenon of

Landau damping is well understood. But it was interesting to find similar effects

for a weakly inhomogeneous plasma. We could not really get a very clear under-

standing of the phenomenon but our effort certainly gave us some insights into the

complex phenomenon.

The case of a noisy plasma is still obscure and a lot of work needs to be done in

that area. We have no doubt explored some avenues but that was just scratching

the surface.

The case of the wave particle interaction was the most intriguing of all. We

brought in nonuniform zero order fields, collisions and unequal flux of particles on

the boundaries of the plasma. We did see fingerprints of Landau damping for this

case also.

37

Bibliography

[1] L.D. Landau, J. Phys. (U.S.S.R.) 10, 25 (1946).

[2] Dwight R. Nicholson, Introduction to Plasma Theory, John Wiley and Sons.

[3] G.J.Morales et al., Phys. Fluids B, Vol. 4, No. 3, March 1992.

[4] E. Kreyszig, Advanced Engineering Mathematics, 8th edition, page 108

38

Appendix A

dist.f

This Fortran 77 program is used to estimate the modified distribution function inthe presence of a landau damped electrostatic wave in a homogeneous plasma.

program dist

integer i,j,N,Nx,Nv

N is the maximum value of Nx and Nv. Nx is the number of divisions of theco-orinate axis and Nv represents the number of divisions of velocity space.

parameter(N=1000)

real fold(N),v(N),vnew(N,N)

fold(N) represents the velocity dependent distribution function before the wave hasset in. This fold(N) is taken to be Maxwellian in our simulation. v(N) representsthe sampled points in velocity space at t=0. vnew(i,j) represents the final velocityof the particle in the ith space bin and jth velocity bin.

real fnewplot(N),vnewplot(N)

fnewplot(N) represents the velocity dependent distribution function at t=tmax.vnewplot(N) represents the sampled points in velocity space at t=tmax.

real tmax,E1,wi,k,vmin,vmax,vwave

tmax is the time till which we run the simulation. E1 is the magnitude of thefirst order sinusoidal electric field. wi is the imaginary part of the frequency ofoscillations of the wave. The rate of damping is determined by wi. This is actuallydependant on the slope of the velocity distribution function at the point where theparticle velocity is equal to the phase velocity of the wave. But in this simulation wehave allowed the user to choose the value of wi. k is the wave number of the wave.vmin and vmax are the minimum and maximum values of the intitial velocities ofthe particles. vwave is the phase velocity of the wave.

39

real vstep,vnewmin,vnewmax,vnewstep

vstep is the step-size in the intial velocity space. vnewmin and vnewmax are theminimum and maximum velocities at t=tmax. vnewstep is the step-size in the finalvelocity space.

external pathfinder

pathfinder is a subroutine used to find the final velocity of a charged particleafter having travelled in a user defined electric field. This in turn uses standardroutines odeint, rkck and rkqs from “Numerical Recipes in Fortran 77”. Thesethree routines have not been included in this code.

real xmin,xman,xstep,x(N)

xmin and xmax are the minimum and maximum values of the intitial positions ofthe particles. xstep is the step-size in the intial co-ordinate space. x(N) representsthe sampled points in the co-ordinate space.

common /pathparameters/ E1,wi,k

These common parameters are used by the subroutine derivs

integer status,system

open(19,FILE="distparameters.dat",STATUS="OLD")

read(19,*) tmax,E1,wi,k,vmin,vmax,vwave,Nv,xmin,xmax,Nx

distparameters.dat is a text file containing the above mentioned values. Thesevalues can be written to this file by using a scilab code.

do i=1,Nv

read(19,*) v(i)

enddo

do i=1,Nx

read(19,*) x(i)

enddo

close(19)

The above two loops are for reading the sample points in velocity and co-ordinatespace from distparameters.dat.

do i=1,Nv

fold(i)=exp(-0.5*((v(i)+vwave)**2))

40

The above intitializes our velocity dependent distribution function to a gaussiancurve.

do j=1,Nx

call pathfinder(tmax,v(i),x(j),vnew(i,j))

enddo

enddo

pathfinder takes as input values tmax, v(i) and x(j) and returns vnew(i,j), whichis the final velocity of a particle at time tmax that started its trajectory at the pointx(j) with an initial velocity of v(i).

vnewmin=vnew(1,1)

vnewmax=vnewmin

do i=1,Nv

do j=1,Nx

if(vnew(i,j).gt.vnewmax) then

vnewmax=vnew(i,j)

elseif(vnew(i,j).lt.vnewmin) then

vnewmin=vnew(i,j)

endif

enddo

enddo

The above two loops are used to find vnewmin and vnewmax. This step is notreally useful when the velocity range is large. But when we are considering a smallregion near the phase velocity of the wave, this can improve accuracy.

vnewstep=(vnewmax-vnewmin)/Nv

do i=1,Nv

vnewplot(i)=vnewmin+i*vnewstep

fnewplot(i)=0

enddo

vnewplot(*) stores the sample points in the new velocity space, i.e. at t=tmax.fnewplot(*) is the modified distribution function. Here it is initialized to zero.

do i=1,Nv

do j=1,Nx

k=(vnew(i,j)-vnewmin)/vnewstep+1

k represents the velocity bin to which the current particle belongs.

41

fnewplot(k)=fnewplot(k)+fold(i)

After having found ’k’, the number of particles in the kth bin are incremented bythe number of particles that were in the ith bin at t=0.

enddo

enddo

We have come to the end of our main program. Now, we just write out theevaluated values to dist.dat which in turn is used by dist.gnu to plot the results.

open(19,FILE="dist.dat",STATUS="UNKNOWN")

do i=1,Nv

write(19,*) v(i),Nx*fold(i)

enddo

write(19,*) ’’

do i=1,Nv

write(19,*) vnewplot(i),fnewplot(i)

enddo

write(19,*) ’’

do i=2,Nv

write(19,*) v(i),vnew(i,1)

enddo

close(19)

status=system("gnuplot<dist.gnu")

status=system("gv dist.ps")

end

subroutine pathfinder(tmax,vo,xo,vfinal)

C driver for routine odeint

INTEGER KMAXX,NMAX,NVAR

PARAMETER (KMAXX=500,NMAX=50,NVAR=2)

INTEGER i,kmax,kount,nbad,nok

REAL dxsav,eps,h1,hmin,x1,x2,x,y,ystart(NVAR)

COMMON /path/ kmax,kount,dxsav,x(KMAXX),y(NMAX,KMAXX)

EXTERNAL derivs,rkqs

As said earlier, pathfinder is a routine that takes in values tmax, vo and xoand returns vfinal. It uses adaptive stepsize Runge-Kutta method of integrationof ordinary differential equations. For storage of intermediate results we have

42

declared the common block path. KMAXX is the size of the array that is usedto store intermediate results. NMAX is the maximum number of variables in thesystem of ordinary differential equations. NVAR is the actual number of variables.In our case these two variables are co-ordinate space and velocity. kmax is themaximum number of intermediate results stored. This can be set to KMAXX.kount is the actual number of stored results. nok and nbad is the number of goodand bad steps taken respectively. dxsav is minimum interval above which the valuesare stored. eps is the desired accuracy. h1 is the guessed first step size. hmin is theminimum allowed stepsize. x1 and x2 are the starting and ending points in time.x stores the intermediate values of time. y(1,*) and y(2,*) store the intermediatevalues of velocity and co-ordiante space respectively. ystart(1) and ystart(2) arethe initial values of velocity and position of a particle respectively. derivs is theroutine that defines the system of differential equations. rkqs is a standard routineused by odeint and can be obtained from “Numerical Recipes in Fortran 77”.

integer status,system

real tmax,vo,xo,vfinal

x1=0

x2=tmax

ystart(1)=vo

ystart(2)=xo

eps=1.0e-4

h1=.1

hmin=0.0

kmax=KMAXX

dxsav=(x2-x1)/KMAXX

call odeint(ystart,NVAR,x1,x2,eps,h1,hmin,

* nok,nbad,derivs,rkqs)

vfinal=y(1,kount)

end

subroutine derivs(x,y,dydx)

real x,y(*),dydx(*)

real e,E1,me,wi,k

parameter(e=1.6e-19,me=9.1e-31)

common /pathparameters/ E1,wi,k

dydx(1)=-(e*E1*exp(-wi*x)/me)*sin(k*y(2))

dydx(2)=y(1)

43

y(1) is velocity and y(2) is position. dydx(1) represents acceleration of the electrondue to the electric field. dydx(2) represents y(1), i.e. velocity. In this simulation,the electric field is a sinusoidal wave with an exponentially decaying amplitude. Thevalues of the E1, wave number, k, and rate of damping, wi, are read from a file,distparameters.dat in the main program and are made available to this subroutinethrough a common block statement. Since the simulation is done in the frame inwhich the wave is at rest, we do not have to specfy the frequency of oscillation ofthe wave.

return

end

44

Appendix B

path.ai.f

This Fortran 77 program is used to simulate the path of particles for the case of aninhomogeneous plasma where the electric field is modeled as a single airy function.

PROGRAM main

integer TSTEP

parameter(TSTEP=200)

TSTEP is the number of time steps from t1 to t2.

integer KMAXX,NMAX,NVAR,Nx,Nv

parameter (KMAXX=5000,NMAX=50,NVAR=2,Nx=100,Nv=100)

KMAXX is maximum value of TSTEP. NMAX is the maximum value of NVAR.NVAR is number of variables in our system of ordinary differential equations. Nxis the number of sample points in the co-ordinate space. Nv is the number of samplepoints in the velocity space.

integer i,j,k

real v1,v2,vstep

v1 is the minimum velocity of a particle at t=0. v2 is the maximum velocity of aparticle at t=0. vstep is the step-size in the velocity space.

real x1,x2,xstep,t,y,vstart(NVAR)

x1 is the minimum value of position of a particle at t=0. x2 is the maximum valueof position of a particle at t=0. xstep is the step-size in the co-ordinate space. tstores the sample points on the time axis. y stores the position and velocity of aparticle at all time instants. vstart stores the initial values of position and velocityof a particle.

45

real t1,t2

t1 represents the starting time of simulations. t2 represents the time till which werun our simulations.

common /path/ t(KMAXX),y(NMAX,KMAXX)

external derivs,rkdumb

derivs is the subroutine where we define our system of differential equations. rk-dumb is a standard subroutines found in “Numerical Recipes in Fortran 77”.

integer turnleft,turnright

turnleft and turnright are used to find out which particles are turning around intheir paths.

integer status,system

open(19,FILE="path.dat",STATUS="UNKNOWN")

open(11,file="tl.dat",status="unknown")

open(12,file="tr.dat",status="unknown")

path.dat is used to store the position and velocity of all particles at all time in-stants. tl.dat and tr.dat contain the intial position and velocity of particles thatturned left and right on their paths respectively. The particles that were found tothe right of their intial positions at some point of time but ended up on the leftside of the initial position are termed to have turned left and vice verca.

x1=0.0

x2=10.0

xstep=(x2-x1)/Nx

v1=-3.5

v2=3.5

vstep=(v2-v1)/Nv

t1=0.

t2=20.

do j=1,Nv

write(*,*) j

do i=1,Nx

turnleft=0

turnright=0

46

vstart(1)=x1+(i-1)*xstep

vstart(2)=v1+(j-1)*vstep

call rkdumb(vstart,NVAR,t1,t2,TSTEP,derivs)

vstart(1) and vstart(2) are the initial position and velocity of the current particle.A call to rkdumb updates the value of y(*,*) to the positions of velocities of thecurrent particle for all time instants. This is then written to a file “path.dat” andis also used to find which particles turnaround.

do k=1,TSTEP

write(19,*) y(1,k),y(2,k),t(k)

if(y(1,k).gt.y(1,k-1)) turnleft=1

if(y(1,k).lt.y(1,k-1)) turnright=1

enddo

The particle can be said to turnleft iff at some point of time it moved to the rightbut eventually ended up on the left of its starting point. Similarly, for turnright.By setting turnleft or turnright to 1, we are marking those particles that qualifyto be categorized as turnleft or turnright respectively. If for a particle, turnleft isequal to 1, means that this particle was found on the right of its intial position atsome point of time. Below, we check if at the end of our simulation time, i.e. afterTSTEP iterations, if the particle was found to the left of its initial position, and ifturnleft is equal to 1, then we write the initial position and velocity of this particleto tl.dat. Similarly, for particles that are termed to have turned right.

if(y(1,TSTEP).lt.y(1,1).and.

* (turnleft.eq.1)) then

write(11,*) vstart(2),vstart(1)

elseif(y(1,TSTEP).gt.y(1,1).and.

* (turnright.eq.1)) then

write(12,*) vstart(2),vstart(1)

endif

write(19,*) ’’

enddo

enddo

close(19)

end

subroutine derivs(t,y,dydt)

real t,y(*),dydt(*)

t represents time. y(1) represents position. y(2) represents velocity.

47

real ai,bi,aip,bip

ai is one of the values returned by the airy function. This is the value that is usedfurther. bi, aip and bip are also the values returned by the airy function but thesevalues are never used any further.

external airy

airy(z,ai,bi,aip,bip) is a standard subroutine used to calculate the airy function.This can be found in “Numerical Recipes in Fortran 77”.

call airy(-y(1),ai,bi,aip,bip)

dydt(1)=y(2)

dydt(2)=-0.4*ai*cos(t)

return

end

48

Appendix C

path.va.f

This Fortran 77 program is used to simulate the path of particles for the case ofan inhomogeneous noisy plasma where the electric field is modeled as the sum ofmultiple shifted and random phased airy functions.

program main

integer KMAXX,NMAX,NVAR,Nx,Nv,Nai

parameter (KMAXX=200,NMAX=50,NVAR=2,Nx=200,Nv=100,Nai=1000)

Nai is the number of sample points of the electric field. In this case, the electricfield is modelled as the sum of shifted and random phased multiple airy functions.

integer NSTEP

parameter(NSTEP=100)

real vstart(NVAR)

integer i,j,k

real x1,x2,xstep,v1,v2,vstep

real t1,t2,t,y

common /path/ t(KMAXX),y(NMAX,KMAXX)

external rkdumb,rk4,derivs,ran0,Efield

real ran0,Efield

ran0 is a standard routine used to generate random numbers. Efield is a functionthat returns the electric field at a given point.

integer status,system

real PI

parameter(PI=3.1415927)

real aiphase(Nai)

aiphase(*) stores the phase information of the various random phased airy func-tions.

49

integer idum

real ai(Nai,Nai),xaimin,xaimax,xaistep,xai(Nai)

ai(*,*) stores the values of the various shifted airy functions. xaimin and xaimaxrepresent the endpoints of our co-ordinate space.

real waimin,waimax,waistep,wai(Nai)

wai(*) stores the frequency of oscillations of the various airy functions. The pointof reflection of an airy wave depends on its frequency.

common /aiparameters/ aiphase,ai,xaimin,

* xaimax,xaistep,xai,wai

The above common parameters are used by the function Efield to calcute theelectric field at any given point.

integer turnleft,turnright

idum=-1

t1=0.

t2=20.

x1=-10.0

x2=10.0

xstep=(x2-x1)/Nx

v1=-10.0

v2=10.0

vstep=(v2-v1)/Nv

xaimin=-50.

xaimax=50.

xaistep=(xaimax-xaimin)/Nai

waimin=0.5

waimax=1.5

waistep=(waimax-waimin)/Nai

do i=1,Nai

aiphase(i)=2.0*PI*ran0(idum)

enddo

50

open(19,file="sairy.dat",status="old")

sairy.dat is a text file containing information about the values of the various shiftedairy functions.

do i=1,Nai

do j=1,Nai

read(19,*) xai(i),ai(i,j)

enddo

enddo

do i=1,Nai

xai(i)=xaimin+(i-1)*xaistep

wai(i)=sqrt(waimin+(i-1)*waistep)

enddo

open(19,file="path.dat",status="unknown")

open(11,file="tl.dat",status="unknown")

open(12,file="tr.dat",status="unknown")

do j=1,Nv

do i=1,Nx

turnleft=0

turnright=0

vstart(1)=x1+(i-1)*xstep

vstart(2)=v1+(j-1)*vstep

call rkdumb(vstart,NVAR,t1,t2,NSTEP,derivs)

do k=1,NSTEP

write(19,*) y(1,k),y(2,k),t(k)

if(y(1,k).gt.y(1,1)) turnleft=1

if(y(1,k).lt.y(1,1)) turnright=1

enddo

write(19,*) ’’

if((y(1,NSTEP).lt.y(1,1)).and.

* (turnleft.eq.1)) then

write(11,*) vstart(2),vstart(1)

elseif((y(1,NSTEP).gt.y(1,1)).and.

51

* (turnright.eq.1)) then

write(12,*) vstart(2),vstart(1)

endif

enddo

enddo

close(19)

end

subroutine derivs(t,y,dydt)

real t,y(*),dydt(*)

real Efield

external Efield

dydt(1)=y(2)

dydt(2)=-Efield(t,y(1))

return

end

function Efield(t,x)

integer Nai,Npol

parameter(Nai=1000,Npol=4)

Npol is the number of points used for doing the polynomial interpolation. We havethe electric field tabulated at known points in space. To calculate the electric fieldat intermediate points, we do polynomial interpolation.

real t,x,Efield

real aiphase(Nai),wai(Nai)

integer i,j

real xaipos,xaipol(Npol)

real ai(Nai,Nai),aipol(Npol)

xaipol(*) and aipol(*) store the values of the co-ordinate space and the field thatare used for getting the interpolated value of the field at the required point.

real xaimin,xaimax,xaistep,xai(Nai)

external polint

real err

polint is a standard routine for doing polynomial interpolation. err is the possibleerror in the interpolated value.

common /aiparameters/ aiphase,ai,xaimin,

* xaimax,xaistep,xai,wai

52

xaipos=int((x-xaimin)/xaistep)+1

xaipos represents the point at which we know the electric field and that is closestto the given point.

if(xaipos.lt.Npol/2) then

Efield=0

return

endif

if(xaipos.gt.(Nai-Npol/2)) then

Efield=0

return

endif

We do not need to bother about points that are on the extreme end of our co-ordinate space. So, we assign them an electric field of 0.

do i=1,Npol

xaipol(i)=xai(xaipos+i-Npol/2)

enddo

do i=1,Npol

aipol(i)=0

do j=1,Nai

aipol(i)=aipol(i)+ai(j,xaipos+i-Npol/2)*

* cos(wai(j)*t+aiphase(j))

enddo

enddo

call polint(xaipol,aipol,Npol,x,Efield,err)

return

end

53

Appendix D

Green’s function technique

Here we discuss the green’s function technique used in getting at Equation 3.30on page 28. This technique applies to differential equations that have the generalform,

y′′+ p(z)y′ + q(z)y = r(z) (D.1)

with arbitrary variable functions p, q and r that are continuous on some intervalI. A prime denotes derivate w.r.t. z. The method gives a particular solution, yp ofequation D.1 on I in the form,

yp(z) = −y1

∫ z

c1

y2r

Wdz + y2

∫ z

c2

y1r

Wdz (D.2)

where y1, y2 form a basis of solutions of the homogeneous equation,

y′′+ p(z)y′+ q(z)y = 0 (D.3)

c1 and c2 are arbitrary constants and W is the Wronskian, given by,

W = y1y2′ − y2y1′ (D.4)

It must however be noted that before applying this technique one should makesure that his/her equation is written in the standard format of equation D.1, withy′′ as the first term; divide by f(z) if it starts with f(z)y′′.

For a proof of this technique please refer to the source [4] from which we havetaken this method .

The equation that we are considering is equation 3.29 on page 28,

∂2E1

∂z2+w2ez

v2eLE1 =

w2en1

v2en0(0)

E0(z) (D.5)

54

We can write this in the standard form of equation D.1 as,

y′′ + zy′ = r(z) (D.6)

where the units have been normalized such that w2e

/v2eL = 1 and

r(z) =w2en1

v2en0(0)

E0(z) (D.7)

We know that the homogeneous equation,

y′′+ zy′ = 0 (D.8)

has two solutions, Ai(-z) and Bi(-z) namely, where Ai(z) and Bi(z) are solutionsof the equation,

y′′ − zy′ = 0 (D.9)

and are plotted in Figures D.1 and D.2 respectively.The Wronskian of Ai and Bi is,

W =1

π(D.10)

Now, we have two make a choice of the limits of the integration and that is themost important part of this problem.

Figure D.1: The airy function, Ai(-z). The frequency of time oscillations, w, ofthis wave is equal to the plasma frequency, we, at z=0. Hence, the wave has acut-off at this point.

55

Figure D.2: The airy function, Bi(-z). The frequency of time oscillations, w, ofthis wave is equal to the plasma frequency, we, at z=0. Hence, the wave has acut-off at this point.

We can see that Bi(-z) is monotonically increasing towards the left. We alsoneed to keep in mind that our modified electric field should have contributionsfrom the entire density function. Thus, we need to choose one among c1 and c2 as−∞ and the other as +∞. The constant associated with Bi(-z) has to be chosento be +∞ because otherwise, our electric field will blow up. Thus, the solution toequation D.6 is,

yp = πB(z)∫ z

−∞A(u)r(u)du+ πA(z)

∫ ∞

zB(u)r(u)du (D.11)

where A(z) and B(z) are Ai(-z) and Bi(-z) respectively and r(u) is given byequation D.7. But it must be noted that when E0 = 0, r(z) = 0, and in thatcase our airy field will also go to zero. But we do not want that to happen. Ourairy field is present even when there is no feedback from the density fluctuations.Thus we add the homogeneous solution, Ai(-z), to the above solution. Thus ourrequired electric field is the one as given by equation 3.30.

E1 = cAi(-z)+Bi(-z)∫ z

−∞Ai(-z)

[ w2en1

v2en0(0)

E0(z)]dz+Ai(-z)

∫ ∞

zBi(-z)

[ w2en1

v2en0(0)

E0(z)]dz

(D.12)

56

Appendix E

feedback.f

This Fortran 77 program is used to simulate the problem of wave particle interac-tion.

program main

integer Nz,Nv,Nzbin,Nvbin,MNp,Np,Nt,SNt,tn

Nz is the number of divisions in the co-ordinate space. Nv is the number of divi-sions in velocity space. Nzbin × Nvbin is the number of baskets in the phase spaceused for statistical purposes. MNp is the maximum number of particles for whichwe calculate the orbits. Np is the actual number of particles. Nt is the number ofdivisions on the time scale SNt determines the relative scale length of the calcula-tions of modification in electric field. We might be interested in situations whenthe electric field varies at a much slower time scale than the density perturbation.SNt=1 implies that the electic field varies as quickly as the density perturbation.tn is the number of steps taken for the calculation of the orbits in each time step.

integer KMAXX,NMAX,NVAR,N

KMAXX is the maximum number of intermediate results of integration that arestored. NMAX is the maximum number of variables in our system of ordinarydifferential equations. NVAR is the actual number of variables in our system,namely, v and z. N is the maximum possible values of Nv, Nz and Nt.

parameter (KMAXX=500,NMAX=20,NVAR=2,N=2000)

parameter (MNp=1000000)

real V0(MNp),Z0(MNp),Zp(MNp),V(MNp)

V0(*) stores the initial velocity of all the particles. V(*) stores the updated ve-locity of all the particles at each time instant. This value is then used in the nextiteration. Z0(*) stores the initial position of all the particles. Zp(*) stores theupdated position of all the particles at each time instant. This value is then usedin the next iteration.

57

integer i,j,k,bz,bv,j0

These are just indices used in our simulation.

real z1,z2,zstep,z(N),v1,v2,vstep,ve,vstart(NVAR),L

z1 and z2 define the boundaries of space. zstep is the length of the smallest divisionin space. v1 and v2 are the minimum and maximum values of the initial velocitiesof the particles. vstep is the length of the smallest division in velocity space. z(*)stores the values of the sampled points in co-ordinate space. ve is the thermalvelocity of the particles. vstart(*) stores the values of the position and velocity ofthe particles for which the new position and velocity at the next time instant is tobe calculated. L is the scale length of the plasma.

real t1,t2,tstep,t(N),tnow,tnext

t1 and t2 are the starting and ending times of our simulation. tstep is the incre-mental step in time. t(*) stores the values of the time instants from t1 to t2. tnowrepresents the time for which we have finished calculating the particle orbits. tnextrepresents the next time instant.

integer N0(N),N1(N),totalgauss(N)

real N1C(N),N1Q(N)

NO(*) is the initial density gradient. This is linear for the current problem. N1(N)is the modified density function. This is the sum of the background gradient and theperturbation due to the electric field. totalgauss(N) is the actual number of particlescorresponding to a particular position. The value stored in N0(i) for some i justrepresents the actual number of particles, i.e. totalgauss(i). N1C(*) and N1Q(*)are the inphase and quadrature components of the density perturbations. N1C(i) =∑t=t2t=t1(N1(i) − N0(i))cos(wt), for all i. N1Q(i) =

∑t=t2t=t1(N1(i) − N0(i))sin(wt),

for all i. For our simulations we have chosen w to be 1.

real energy(N),En(N),E(N)

energy(*) stores the total kinetic energy of all the particles at all time instants.En(i) stores the total kinetic energy of the particles present in between positions iand i+1 at a particular time instant. E(*) stores the z-dependant electric field forall points in space at a particular time instant.

real zbinstep,vbinstep

integer N1bin(N,N)

real N1binC(N,N),N1binQ(N,N),Vbin(N,N),Enbin(N,N)

58

zbinstep is the step size of one bin in space. vbinstep is the step size of one binin velocity space. N1bin(*,*) is the number of particles in a particular bin at aparticular time. N1binC(*,*) and N1binQ(*,*) are the inphase and quadraturecomponents of N1bin(*,*). These are defined in a manner similar to N1C(*) andN1Q(*). Vbin(*,*) is the drift velosity of a bin at a particular time. En(*,*) isthe total kinetic energy of all particles in a bin at a particular time.

external ran0,derivs,rkdumb,distf

real ran0

integer distf

ran0 is a random number generator. derivs is a routine that is used to defineour system of ordinary differential equations, i.e., (̇z) = v and v̇ = a. It is thisa that is the force due to the electric field. rkdumb is a standard routine used tosolve a system of ordinary differential equations. We could have better codes butthat would make the simulation extremely slow. distf is a function that is usedto actual number of particles corresponding to a particular valus of initial velocityand position. ran0 and rkdumb are standard routines and can be obtained from“Numerical Recipes in Fortran 77”.

real ttemp(KMAXX),ytemp(NMAX,KMAXX)

ttemp(*) stores the intermediate values of time instants used by rkdumb to per-form the integration. ytemp(1,*) and ytemp(2,*) store the intermediate values ofposition and velocity as calculated by rkdumb.ytemp(1,tn) and ytemp(2,tn) are thefinal values.

real fzv

real p,pr

integer idum

fzv is just a temporary variable and is used to reduce the number of function callsto the function distf. p and pr are used to model the collisions. p determinesthe rate of collisions. A higher p implies more collisions. pr is defined at theplace where it is used in the code. idum is the seed value for the random numbergenerator.

real ai(N),bi(N),aip(N),bip(N)

ai(*) and bi(*) are the inverted versions of the two airy functions Ai and Bi.aip(*) and bip(*) are their derivatives. In our simulation we have used only ai(*).

integer Npos

integer pathpos(MNp)

59

Npos is the number of points for which we want to keep track of their entire orbitsin phase space. pathpos(*) stores the indicies of the orbits that we want to keeptrack of.

integer Afac(MNp),Aplus,Aminus

When a particle reaches one of the boundaries of our plasma, z1 or z2, we move itto the other extreme and make its velocity equal to its initial velocity. But due tothe gradient in the plasma, for each particle that leaves the boundary, the numberof particles entering at z1 will be more than the number of particles entering at z2.To model this we define an array Afac(*). Afac(i) stores the relative value of thenumber of actual particles corresponding to the particle with index i. Aplus andAminus define the relative number of particles entering at the two end points inspace.

real phi,E0field

external phi,E0field,Efield

phi and E0field are functions that return the zero order potential and electric fieldrespectively.

−eE0

m=−v2

e

L− z (E.1)

Efield is a routine that calculates the modified electric field on each iteration. Thisincludes only the airy function and the green’s function and not the zero orderfield.

∂2E1

∂z2+w2ez

v2eLE1 =

w2en1

v2en0(0)

E0 (E.2)

common /path/ ttemp,ytemp

common /density/ N1,N0

common /calce/ E

common /airydat/ z,ai,bi,aip,bip

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

open(9,file="feedparameters.dat",status="old")

open(10,file="airy.dat",status="old")

open(11,file="zv.dat",status="unknown")

open(12,file="ncq.dat",status="unknown")

open(14,file="path.dat",status="unknown")

open(15,file="bin.dat",status="unknown")

open(17,file="Enz.dat",status="unknown")

open(18,file="En.dat",status="unknown")

open(19,file="N.dat",status="unknown")

open(20,file="E.dat",status="unknown")

60

feedparameters.dat contains the initial conditions and and various parameters forthe simulation to run. airy.dat contains the values of the inverted airy function.zv.dat is the file to which can be used as a feedparameters.dat if we want to run thesimulation again from the time at which we finished last time. ncq.dat stores theinphase and quadrature components of the density perturbations. path.dat storesthe phase space orbits of Npos number of particles. bin.dat storess the valuesof N1bin(*,*),N1binC(*,*),N1binQ(*,*),Vbin(*,*) and Enbin(*,*) at time t=t2Enz.dat stores the values of En(*) at all time instants. En.dat stores the values ofenergy(*) at all time instants. N.dat stores the density profile at all time instants.E.dat contains the z-dependant modified airy electric field at all time instants.

read(9,*) p,Aplus,Aminus,z1,z2,Nz,Nzbin,v1,v2,Nv,Nvbin,ve,

* t1,t2,Nt,SNt,tn,L,Np,Npos

do i=1,Np

read(9,*) Z0(i),V0(i)

enddo

do i=1,Nz

read(9,*) N0(i),totalgauss(i)

enddo

close(9)

do i=1,Nz

read(10,*) z(i),ai(i),bi(i),aip(i),bip(i)

call Efield()

write(20,*) z(i),E(i)

write(19,*) z(i),N0(i)

enddo

close(10)

write(20,*) ’’

write(19,*) ’’

The above is used to read the values of the various parameters from the filesfeedparameters.dat and airy.dat.

idum=1

zstep=(z2-z1)/Nz

zbinstep=(z2-z1)/Nzbin

vbinstep=(v2-v1)/Nvbin

tstep=(t2-t1)/Nt

61

do i=1,Np

Zp(i)=Z0(i)

V(i)=V0(i)

if((V0(i)**2).lt.(2*phi(Z0(i),z2))) then

Afac(i)=Aplus

elseif(V0(i).lt.0) then

Afac(i)=Aminus

else

Afac(i)=Aplus

endif

enddo

do i=1,Nz

N1(i)=0

N1C(i)=0

N1Q(i)=0

enddo

do i=1,Nzbin

do j=1,Nvbin

N1binC(i,j)=0

N1binQ(i,j)=0

enddo

enddo

The above loops initialize the values of the various variables.

do i=1,Npos

pathpos(i)=int(Np*ran0(idum))+1

enddo

The above loop randomly selects Npos orbits that will be written to the file path.dat.

do k=1,Nt/SNt

do i=1,SNt

t((k-1)*SNt+i)=t1+((k-1)*SNt+i-1)*tstep

do j=1,Np

vstart(1)=Zp(j)

62

vstart(2)=V(j)

tnow=t((k-1)*SNt+i)

tnext=tnow+tstep

call rkdumb(vstart,NVAR,tnow,

* tnext,tn,derivs)

if(ytemp(1,tn).gt.z2) then

Zp(j)=z1

V(j)=abs(V0(j))

Afac(j)=Aplus

elseif(ytemp(1,tn).lt.z1) then

Zp(j)=z2

V(j)=-abs(V0(j))

Afac(j)=Aminus

else

Zp(j)=ytemp(1,tn)

V(j)=ytemp(2,tn)

endif

This loop calculates the orbits of all the particles at each time instant. The ifstatement above is used to handle those particles that move out of the boundariesof the plasma, z1 and z2.

pr=ran0(idum)

if(pr.lt.(p*abs(V(j)))) V(j)=-V(j)

The above two lines are used to model the collisions. The velocity of the jth particleis swapped with a probability of p|V (j)|.

enddo

enddo

do i=1,Npos

write(14,*) Zp(pathpos(i)),V(pathpos(i))

enddo

write(14,*) ’’

do i=1,Nz

N1(i)=0

En(i)=0

enddo

do i=1,Nzbin

63

do j=1,Nvbin+2

N1bin(i,j)=0

enddo

enddo

energy(k)=0

do i=1,Np

j=int((Zp(i)-z1)/zstep)+1

j0=int((Z0(i)-z1)/zstep)+1

j maps the current location of a particle to the nearest sampled point in position.j0 maps the initial location of a particle to the nearest sampled point in position.

fzv=distf(V0(i),Afac(i),N0(j0),totalgauss(j0))

fzv stores the value returned by the function distf. This is the actual number ofparticles represented by the orbit with index i. This number is then added to thejth bin to get the values of density and energy as a function of z.

N1(j)=N1(j)+fzv

En(j)=En(j)+(V(i)**2)*fzv

energy(k)=energy(k)+(V(i)**2)*fzv

bz=(Zp(i)-z1)/zbinstep+1

bv=(V(i)-v1)/vbinstep+1

if(bv.lt.1) then

bv=1

elseif(bv.gt.(Nvbin)) then

bv=Nvbin

endif

N1bin(bz,bv)=N1bin(bz,bv)+fzv

bz and bv are the indices of a particular bin in phase space. N1bin(bz,bv) is thenincreased by the value fzv. Calculating N1bin gives us good insight into the relativedistribution of particle density in phase space. We can then calculate the inphaseand quadrature components of this quantity to see the wave features in density.

enddo

if((k.gt.300).and.(k.le.991)) then

do i=1,Nzbin

do j=1,Nvbin+2

N1binC(i,j)=N1binC(i,j)+

64

* float(N1bin(i,j))*cos(tnext)

N1binQ(i,j)=N1binQ(i,j)+

* float(N1bin(i,j))*sin(tnext)

enddo

enddo

endif

N1binC and N1binQ are the inphase and quadrature components of N1bin, whichin turn stores the density of particles in different bins in phase space.

write(18,*) k,energy(k)

write(*,*) ’energy=’,k,energy(k)

do i=1,Nz

N1(i)=int(float(N1(i))/1000)

if((k.gt.300).and.(k.le.991)) then

N1C(i)=N1C(i)+float(N1(i))*cos(tnext)

N1Q(i)=N1Q(i)+float(N1(i))*sin(tnext)

endif

write(19,*) z(i),N1(i)

write(17,*) z(i),En(i)

enddo

write(19,*) ’’

write(17,*) ’’

N1C and N1Q are the inphase and quadrature components of the z-dependent den-sity profile. These quantities also have wave like features. We have chosen 300and 991 for two reasons. One is to remove the effects of early transients and theother is to remove truncation effects. We should average the quantities over aninteger number of wavelengths.

call Efield()

do i=1,Nz

write(20,*) z1+(i-1)*zstep,E(i)

enddo

write(20,*) ’’

enddo

vstep=(v2-v1)/Nv

65

do i=1,Np

j0=int((Z0(i)-z1)/zstep)+1

bz=(Zp(i)-z1)/zbinstep+1

bv=(V(i)-v1)/vbinstep+2

if(bv.lt.1) then

bv=1

elseif(bv.gt.(Nvbin+2)) then

bv=Nvbin+2

endif

fzv=distf(V0(i),Afac(i),N0(j0),totalgauss(j0))

Vbin(bz,bv)=Vbin(bz,bv)+V(i)*fzv

Enbin(bz,bv)=Enbin(bz,bv)+(V(i)**2)*fzv

enddo

Vbin(*,*) stores the velocity of each bin in phase space. Similarly, Enbin(*,*)stores the total energy of the particles in each bin in phase space.

do i=1,Nzbin

do j=1,Nvbin+2

write(15,*) z1+(i-1)*zbinstep,v1+(j-1)*vbinstep,

* N1bin(i,j),N1binC(i,j),N1binQ(i,j),

* Vbin(i,j),Enbin(i,j)

enddo

write(15,*) ’’

enddo

do i=1,Nz

write(12,*) z(i),N1C(i),N1Q(i)

enddo

write(11,*) p,Aplus,Aminus,z1,z2,Nz,Nzbin,v1,v2,Nv,Nvbin,ve,

* tnext,tnext+100.0,Nt,SNt,tn,L,Np,Npos

do i=1,Np

write(11,*) Zp(i),V(i)

enddo

do i=1,Nz

write(11,*) N0(i),totalgauss(i)

enddo

end

subroutine derivs(t,y,dydt)

66

derivs is the routine that defines our system of differential equations, namely,dz/dt = v and dv

/dt = a, where a is the acceleration of electrons due the electric

field. So, it basically defines a.

integer N,Nz,pol

parameter (N=2000,pol=6)

N is the maximum allowed value of Nz and Nz is the number of sample pointsin the co-ordinate space. In our code, we recalculate the electric field at Nz pointsin each iteration in the main program. To get the field strength at intermediatepoints, we use polyomial interpolation. pol is the number of points we use to getthe interpolated value of field strength at a particular point. This must be an evenvalue for this code to give correct results.

real Epol(pol),Zpol(pol)

Epol(*) and Zpol(*) store the values of the field strength and z-coordinates of thepoints used for interpolation.

integer i,j

real t,y(*),dydt(*)

t represents time. y(1) represents z. dydt(1) and y(2) represent v. dydt(2) repre-sents acceleration due to the electric field force.

real E(N)

E(*) stores the values of the electric field strength at Nz sample points in space.

real z1,z2,zstep,z

external polint,E0field

real E0field

polint is a standard routine used for doing polynomial interpolation. E0field isa routine that returns the value of the zero order electric field at a given point inspace.

real Efz,err

Efz is the interpolated value of the first order field strength at a particular pointin space. err is the error in the interpolation estimate.

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

common /calce/ E

z=y(1)

67

i=int((z-z1)/zstep)+1

if(i.lt.pol) then

do j=1,pol

Epol(j)=E(i+j)

Zpol(j)=z1+(i+j)*zstep

enddo

elseif(i.gt.(Nz-pol)) then

do j=1,pol

Epol(j)=E(i-j)

Zpol(j)=z1+(i-j)*zstep

enddo

else

do j=1,pol

Epol(j)=E(i-j+pol/2)

Zpol(j)=z1+(i-j+pol/2-1)*zstep

enddo

endif

call polint(Zpol,Epol,pol,z,Efz,err)

dydt(1)=y(2)

dydt(2)=E0field(y(1))+Efz*cos(t)

return

end

subroutine Efield()

integer N,Nz

parameter (N=2000)

real E(N)

integer i,j

external qromb,ain1,bin1

real ain1,bin1

real term1(N),term2(N),term3(N),term4(N),term5(N)

real term6(N),term7(N)

The first order electric field is given by the equation,

E1 = c(iAi) + iBi∫ z

−∞(iAi)

[ w2en1

v2en0(0)

E0(z)]dz + iAi

∫ ∞

z(iBi)

[ w2en1

v2en0(0)

E0(z)]dz

(E.3)qromb is used to do the integration. ain1 is a routine that returns the integrand ofthe 2nd term. bin1 is a routine that returns the integrand of the 3rd term. term1,

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term2, till term7 are temporary variables used for storing intermediate values ofvarious terms in the above equation. term3 is the value of the integration term inthe 2nd term. term5 is term3 times the value of iBi. term4 is the value of theintegration term in the 3rd term. term6 is term4 times the value of iAi. term7 isthe sum of term4 and term6.

real ai(N),bi(N),aip(N),bip(N)

real z1,z2,zstep,z(N)

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

common /airydat/ z,ai,bi,aip,bip

common /calce/ E

zstep=(z2-z1)/Nz

z(1)=z1

term1(1)=0

do i=2,Nz

call qromb(ain1,z(i-1),z(i),term1(i))

term3(i)=term1(i-1)+term1(i)

enddo

term2(Nz)=0

do i=Nz-1,1,-1

call qromb(bin1,z(i),z(i+1),term2(i))

term4(i)=term2(i)+term2(i+1)

enddo

term3(1)=term1(1)

term4(Nz)=term2(Nz)

do i=1,Nz

term5(i)=term3(i)*bi(i)

term6(i)=term4(i)*ai(i)

term7(i)=term5(i)+term6(i)

E(i)=-0.012*ai(i)+term7(i)/(2000.0*(ve**2))

enddo

return

end

function ain1(z)

integer N,Nz

parameter (N=2000)

real z

real ain1,E0field

external airy,E0field

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integer N1(N),N0(N)

integer pol

parameter (pol=6)

real E0

real Npol(pol)

real Nfin

real Zpol(pol)

common /density/ N1,N0

integer i,j

real temp

real z1,z2,zstep

real ai(N),bi(N),aip(N),bip(N),aipol(pol)

common /airydat/ zt(N),ai,bi,aip,bip

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

i=int((z-z1)/zstep)+1

if(i.lt.pol) then

do j=1,pol

aipol(j)=ai(i+j)

Npol(j)=N1(i+j)-N0(i+j)

Zpol(j)=z1+(i+j)*zstep

enddo

elseif(i.gt.(200-pol)) then

do j=1,pol

aipol(j)=ai(i-j)

Npol(j)=N1(i-j)-N0(i-j)

Zpol(j)=z1+(i-j)*zstep

enddo

else

do j=1,pol

aipol(j)=ai(i-j+pol/2)

Npol(j)=N1(i-j+pol/2)-N0(i-j+pol/2)

Zpol(j)=z1+(i-j+pol/2-1)*zstep

enddo

endif

call polint(Zpol,Npol,pol,z,Nfin,temp)

call polint(Zpol,aipol,pol,z,ain1,temp)

ain1=ain1*Nfin*E0field(z)

return

end

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function bin1(z)

integer N,Nz

parameter (N=2000)

real bin1

real z

real E0field

external airy,spline,splint,E0field

real sp1(2),sp2(2),sp3(2)

integer N1(N),N0(N)

integer pol

parameter (pol=6)

real E0

real Npol(pol)

real Nfin

real Zpol(pol)

common /density/ N1,N0

integer i,j

real temp

real z1,z2,zstep

real ai(N),bi(N),aip(N),bip(N),bipol(pol)

common /airydat/ zt(N),ai,bi,aip,bip

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

i=int((z-z1)/zstep)+1

if(i.lt.pol) then

do j=1,pol

bipol(j)=ai(i+j)

Npol(j)=N1(i+j)-N0(i+j)

Zpol(j)=z1+(i+j)*zstep

enddo

elseif(i.gt.(Nz-pol)) then

do j=1,pol

bipol(j)=bi(i-j)

Npol(j)=N1(i-j)-N0(i-j)

Zpol(j)=z1+(i-j)*zstep

enddo

else

do j=1,pol

bipol(j)=bi(i-j+pol/2)

Npol(j)=N1(i-j+pol/2)-N0(i-j+pol/2)

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Zpol(j)=z1+(i-j+pol/2-1)*zstep

enddo

endif

call polint(Zpol,Npol,pol,z,Nfin,temp)

call polint(Zpol,bipol,pol,z,bin1,temp)

bin1=bin1*Nfin*E0field(z)

return

end

function distf(v,c,n,tg)

real v

integer distf,dist

external dist

integer n,tg

integer c

real ve

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

distf=c*int(1e5*exp(-0.5*(v/ve)**2)*float(n)/float(tg))

return

end

function phi(zstart,zend)

real zstart,zend

real z1,z2,ve,L

real phi

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

phi=(ve**2)*log((L-zstart)/(L-zend))

return

end

function E0field(z)

real z,E0field

real z1,z2,ve,L

common /feedparameters/ z1,z2,zstep,ve,L,Nz,Nv

E0field=-(ve**2)/(L-z)

return

end

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VITA

Name : Kushal Kumar Shah

Address : c/o- Maliram BaijnathMotiganj BazarBalasore - 756003OrissaPh: (06782) 262601(R), 262137(O)Email: atmabodha@gmail.com

D.O.B : 3rd November 1981

Father : Sri Kishan Shah

Mother : Srimati Asha Shah

1998 : 10th from St. Vincent’s Convent School, Balasore, Orissa

2000 : 12th from St. James’ School, Kolkata

2005 : Bachelor of Technology in Electrical Engineering from IIT Madras

73